In this thesis the Brill-Noether loci W of rank 2 stable vector bundles of canonical determinant over an algebraic curve are studied. We analyse three conjectures on the nonexistence, dimension and smoothness of W, collectively known as the Brill- Noether conditions. The local structure of W is described by a symmetric Petri map; assuming that W ≠ Ø, the injectivity of this map ensures the dimension and smoothness conditions we are aiming for. The nonemptiness of W is shown by constructing the appropriate bundles from extensions of line bundles. In a similar vein the nonexistence conjecture is addressed by showing that certain bundles are extensions of line bundles that are prohibited on the curve. Finally, subject to an assumption, the Petri map is shown to be injective for genus ≤10; which allows us to prove that the Brill-Noether conditions hold for genus ≤10, improving on the genus ≤7 results of Bertram-Feinberg [4] .